Question

Three identical capacitors $C_1,C_2$ and $C_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${\in }_r$. The cell electromotive force (emf) $V_o=8V$. First, the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$. The value of ${\in }_r$ is

Answer 1

When $S_1$ is closed and $S_2$ is open.
Initial charge on $C_3=C{V}_o=1\times 8=8\mu C$
When $S_2$ is closed and $S_1$ is open.
Charge will be distributed equally across $C_1$ and $C_2$ i.e $3\mu C$
Applying Kirchhoff's voltage law

$\frac{5}{1}-\frac{3}{{\in }_r}-\frac{3}{1}=0$ $\Rightarrow$ $\frac{3}{{\in }_r}=2$
$\Rightarrow$ ${\in }_r=1.50$